Nonlinear Optics

Nonlinear Optics Lab

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Nonlinear Optics

(비선형 광학)

담당 교수 : 오 차 환

교 재 : A. Yariv, Optical Electronics in Modern Communications, 5^th Ed., Oxford university Press, 1997

부교재 : R. W. Boyd, Nonlinear Optics, Academic Press, 1992

A. Yariv, P. Yeh, Optical waves in Crystals, John Wiley & Sons, 1984

Nonlinear Optics Lab

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Chapter 1. Electromagnetic Theory

1.0 Introduction

Propagation of plane, single-frequency electromagnetic waves in - Homogeneous isotropic media - Anisotropic crystal media

1.1 Complex-Function Formalism

Expression for the sinusoidally varying time functions ;

], A

Re[

] 2 [

| A ) |

cos(

| A

| )

( t t

e

e

e

a     





i a

e

| A

| A

where



Typical expression ;

a ( t )  A e



 ??

Nonlinear Optics Lab

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Distinction between the real and complex forms

1)

a t i e

dt

d ( )    | A | sin(  t  

)   A

2)

[cos( 2 ) cos( )]

2

| B

||

A ) |

( )

( t b t t

a          

) 2

|

B

||

A

| e



* Time averaging of sinusoidal products

) 2 cos(

| B

||

A ) |

cos(

| B

| ) cos(

| A 1 |

) ( ) (

0

b a

b T

a

t dt

T t t

b t

a           

 

*) AB 2 Re(

 1

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1.2 Considerations of Energy and Power in Electromagnetic Field

Maxwell’s curl equations (in MKS units) ;

 t

 





 d

i

h  t

 





 b

e [ , ] d  

e  p b  

(h  m) p

e e e i

e h

e t  t

 



 

 









 ( )

2



m h

h h e

h t  t

 



 

 









( )

2 



Vector identity ;

  ( A  B)  B    A  A    B

t t

t 

 

 

 

 



 



   



 









 m

p h e h h e

e i

e h) e

-   

2

( 2

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Divergence theorem ;

 ^{  }  ^

s v

da dv A n A)

( v s

n

t dv t

da t

dv

v

 

 ^{        } _ ^{ } _ ^{ } _ ^{    } _ ^{   } _ ^{  } _ ^ _ ^

 m

p h e h

h e

e i

e n

h) e h)

e   

2 ( 2

(

: Poynting theorem

Total power flow into the volume

bounded by s

Power expended by the field on the moving charges

Rate of increase of the vacuum electromagnetic

stored energy

Power per unit volume expended by

the field on electric and magnetic dipoles

Nonlinear Optics Lab

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Dipolar dissipation in harmonic fields

The average power per unit volume expended by the field on the medium electric polarization ;

t



 p

volume e power

Assume, field and polarization are parallel to each other

] Re[

)

(t Ee^{i t}

e  ^ p(t)Re[Pe^it], ^where P





) Re(

| 2 |

*]

2Re[

1 volume

power ₂

0

0 e e

t iω t

iω iωω i EE E i

E     

 Re[ e ]Re[ e ]

Put, _e_e'i_e"

0 2

| 2 |

"

volume power

e

E



 

)

* 2



Re( i

E

E

   

: Isotropic media

: Anisotropic media

Nonlinear Optics Lab

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Ex) single localized electric dipole, μ(ex) power

DF t





 

e

Let, position of electron : electric field :

)

0cos( t e

x

x





E

e_x ₀cos



power

DF ₀cos [ ex₀cos( t _e)] ex₀E₀cos tsin( t _e) t t

E







  





 





1) :

2

_e power

DF 





2) :

2

_e power

DF 





: The dipole(electron) continually loses power to the field : The field continually gives power to the dipole

 Power exchange between the field and medium via dipole interaction

Nonlinear Optics Lab

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1.3 Wave Propagation in Isotropic Media

Electromagnetic plane wave propagating along the z-axis in homogeneous, isotropic,

and lossless media (





Put, ee_xu_x, hh_yu_y

t ε e z

h t

h z

ex y y _x



 

 





 

 

  , ² ₂ ² , ² ₂ ² ₂

t ε h z

h t

e z

ex x y y



 







 



  

General solutions : e_x(z,t)E_x^e^i(tkz)E_x^e^i(tkz), ^{( , ) 1}





kz t i x

y z t E e E e

h  ^{  }  ^{  }



* Phase velocity :

n c ε

c k  1  ⁰





* wavelength :



  ^c

k 2 2 



* Relative amplitude :



 

 ^

 ^

 ^x , where

y

H E

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Power flow in harmonic fields

Intensity (average power per unit area carried in the propagation direction by a wave) :

*]

2Re[

| 1

|e h e_xh_y E_xH_y I   



(1.3-17)  _

 

2

|

| 2

| ] |

* ) (

* ) [(

] [

2 Re

1 _{ }  _  _  _{ }  ^{ 2} ^{ 2}





 _x ^ikz _x ^ikz _x ^ikz _x ^ikz E^x E^x

e E

e E

e E e

E I

Electromagnetic energy density :

*}

2Re{

1

*} 2 2Re{

1 2 2

2

2 2

y y x

x y

x h E E H H

V e







_{    }

E 

(1.3-17)  {| | | | }

2 1 2

2

2 2

2

2     



 e_x h_y E_x E_x V



 E

For positive traveling wave : E E c V

I

x

x  



 _{ }







| 1 2| /

| 2 |

1 /

2 2

E



2

1 _{ 2 2}





I c



Nonlinear Optics Lab

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1.4 Wave Propagation in Crystals-The Index Ellipsoid

In general, the induced polarization is related to the electric field as























zz zy

zx

yz yy

yx

xz xy

xx

E,























₀ where

P

: electric susceptibility tensor

























) (

) (

) (

' ' 3 ' 3 ' ' 2 ' 3 ' ' 1 ' 3 0 '

' ' 3 ' 2 ' ' 2 ' 2 ' ' 1 ' 2 0 '

' ' 3 ' 1 ' ' 2 ' 1 ' ' 1 ' 1 0 '

z y

x z

z y

x y

z y

x x

E E

E P

E E

E P

E E

E P

























If we choose the principal axes, (Diagonalization)













z z

y y

x x

E P

E P

E P

33 0

22 0

11 0













z y x ,,













z z

y y

x x

E D

E D

E D

33 22 11

























) 1

(

) 1

(

) 1

(

33 0

33

22 0

22

11 0

11

where



















/0

 n

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) (

)

( ^{t k r} , ^{i t k r}

i e

e ^

 

  

 

E₀ ^ H H₀ ^ E

Secular equation

For a monochromatic plane wave ;

From Maxwell’s curl equations,

2 2

t

 









 E

E 

0 )

(   ² 

 k E

  

In principal coordinate,



















z y x

ε ε ε

0 0

0 0

0 0



0

2 2 2

2 2 2

2 2 2













































z y x

y x z y

z x

z

z y z

x y x

y

z x y

x z

y x

E E E

k ε k

k k k

k

k k k

ε k k

k

k k k

k k

ε k













Nonlinear Optics Lab

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Simple example ( k_xk, k_yk_z0^{) :} wave propagating along the x-axis

 



 













0 )

(

0 )

(

0

2 2

2 2

2

z z

y y

x x

E ε k

E ε k

ε E















 











0 ,

0 ,

and and

y z

z y

ε E k

ε E k









For nontrivial solution to exist, Det=0 ;

0

2 2 2

2 2 2

2 2 2















y x z y

z x

z

z y z

x y x

y

z x y

x z

y x

k ε k

k k k

k

k k k

ε k k

k

k k k

k k

ε k













Nonlinear Optics Lab

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kz

kx

ky

c n_z/ c

n_z/

c n_x/

c n_x/

c n_y/

c n_y/

Normal surface

Optic axis

Simple example ( k_z0

, determinant equation 

2 0

2 2

2 2 2

2 2 1

2 2

3 











 











  



 















  



 















   



 





y x x

y y

x k k k

c k n

c k n

c k

n   

)

2 2 3

2 



 





 c

k n

k_{x y}  : circle

1

1 2

2

2 



 







 





c n

k

c n

k_{x y}





: ellipse c s

k^_n ˆ

Nonlinear Optics Lab

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Index ellipsoid

The surface of constant energy density in D space :

e z

z y

y x

x D D U

D 2

2 2

2   







Energy density :

j i ij

e E E

U



2

1

r Ue

D/ 2

/ 1 /

/ ₀

2

0 2

0

2   













z y

x or ₂ 1

2 2

2 2

2   

z y

x n

z n

y n

x : Index ellipsoid

Nonlinear Optics Lab

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Classification of anisotropic media

1) Isotropic : n_xn_yn_z

ex) CdTe, NaCl, Diamond, GaAs, Glass, …

2) Uniaxial : n_xn_yn_z

(1) Positive uniaxial : n_zn_x

ex) Ice, Quartz, ZnS, … (2) Negative uniaxial : n_zn_x

ex) KDP, ADP, LiIO₃, LiNbO₃, BBO, …

) :

, :

(n_zn_e extraordinary n_xn₀ ordinary  Fast/Slow axis

3) Biaxial : n_xn_yn_z

ex) LBO, Mica, NaNO₂, …

Nonlinear Optics Lab

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Example of index ellipsoid (positive uniaxial)

2 1

2 2

0 2

2  

ne

z n

y x

) sin ,

cos ,

0

( n_e





sˆ

x

y z



) 0 , ,

0

( n₀

) 0 , 0 , (n₀

) , 0 , 0

( n_e

B A

0

propagation direction

Nonlinear Optics Lab

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Intersection of the index ellipsoid

sˆ

y z



A

0



2 2

2( ) z y

n_e



2 1

2 2

0

2  

ne

z n

y









( _e

e y n

n

z 

) ( 1 sin

cos

2 2

2 2

0 2







e

e n

n

n  



Birefringence : |n_e(



0 0

0| 0, | (90) | )

0 (

|n_e n  n_e n n_en

Nonlinear Optics Lab

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Normal index surface

: The surface in which the distance of a given point from the origin is equal to the index of refraction of a wave propagating along this direction.

1) Positive uniaxial (n_e>n_o)

z

y ne

n0

n0

2) negative uniaxial (n_e<n_o) z

y n0

ne

n0

3) biaxial ( ) z

y ny

nx n_z

z y

x n n

n  

Nonlinear Optics Lab

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1.5 Jones Calculus and Its Application in Optical Systems with Birefringence Crystals

Jones Calculus (1940, R.C. Jones) :

- The state of polarization is represented by a two-component vector - Each optical element is represented by a 2 x 2 matrix.

- The overall transfer matrix for the whole system is obtained by multiplying all the individual element matrices.

- The polarization state of the transmitted light is computed by multiplying the vector representing the input beam by the overall matrix.

Examples)

- Polarization state :

- Linear polarizer (horizontal) :

- Relative phase changer :



 



 

y x

V V V



 





0 0

0 1



 





 _y

x

i i

e e





0

0

Report) matrix expressions

- Linear polarizers (horizontal, vertical) - Phase retarder

- Quarter wave plate (fast horizontel, vertical) - Half wave plate

Nonlinear Optics Lab

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Retardation plate (wave plate)

: Polarization-state converter (transformer)

Polarization state of incident beam :



 





y x

V

V V where, _{V ,x Vy} : complex field amplitudes

along x and y

s, f axes components :



 



 



 







 





 



 





y x y

x

V R V

V V V

V ( )

cos sin

sin cos

f

s 









Polarization state of the emerging beam :



































 





 



 



















f s

f s

f s

exp 0

0 exp

V V c l

in c l

in V

V





Nonlinear Optics Lab

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Define,

- Difference of the phase delays :

c n l

n 

) ( _s  _f



 - Mean absolute phase change :

c n l

n 

 ( )

2 1

f s 





 

























 







  _

 



f s 2

2

f s

0

0

V V e

e e V

V

i i

i

Polarization state of the emerging beam in the xy coordinate system :



 







 



 



 

 



 









f s

cos sin

sin cos

V V V

V

y x







 

 



 





 







 

y x y

x

V R V

W V R

V (





cos , sin

sin ) cos

( 

 





 







 

R _



 



 ^  ^ _

2 / 2

/

0 0

0

i i

i

e e e

W ^

where,

Nonlinear Optics Lab

.

Transfer matrix for a retardation plate (wave plate)



















2 )

2 / ( 2

) 2 / ( 2

) 2 / ( 2

) 2 / (

0

cos sin

) 2 2 sin(

sin

) 2 2 sin(

sin sin

cos

) ( )

( )

, (













 



 

 









i i

i i

e e

i

i e

e

R W R

W W

1

W

Transfer matrix is a unitary ( ) W

: Physical properties are invariant under unitary transformation

=> If the polarization states of two beams are mutually orthogonal, they will remain orthogonal after passing through an arbitrary wave plate.



Nonlinear Optics Lab

.

Ex) Half wave plate



 











 1

: 0 beam incident

, 4 /

,  V



4 / cos 4

/ sin )

2 / 2sin(

sin

) 2 / 2sin(

sin 4

/ sin 4

/ cos

2 ) 2 / ( 2

) 2 / ( 2

) 2 / ( 2

) 2 / (





 

 













i i

i i

e e

i

i e

e W







 







11) 5 . 1 (

0 0

i i



 



 



 





 







 







 







 

 0

1 0

1 0 0

' 0 i i

i

V i : x-polarized beam

Report : Problem 1.7

Nonlinear Optics Lab

.

Ex) Quarter wave plate



 











 1

: 0 beam incident

, 4 / ,

2

/  V





11) 5 . 1

( 1

1 2 1

i W i



 



 



 





 



 



 







 







 

 i

i i i

V i 1

1 2 2 1 1

0 1 1 2 ' 1

: left circularly polarized beam

: y-pol.



 











 0

: 1 beam incident

, 4 / ,

2

/  V





 





 



 







 







 

 i i

V i 1

2 1 0

1 1 1 2 ' 1

: right circularly polarized beam

: x-pol.

Nonlinear Optics Lab

.

Intensity transmission

In many cases, we need to determine the transmitted intensity, since the combination of retardation plates and polarizers is often used to control or modulate the transmitted optical intensity.

Incident beam intensity :



 



 

y x

V

V V ^{2 2}

y

x

V

V

I    

 V V

Output beam intensity : 



 







 



y x

V V V

' 2 ' 2

' V

V

I  



Transmissivity :

2 2 2 2

y x

y x

V V

V V



 



Nonlinear Optics Lab

.

Ex) A birefringent plate sandwiched between parallel polarizers , ) (

2 _{e o}

 n n ^d



 /4











 





 























 

 





 







cos2 0 1

0

cos2 sin2

sin2 cos2

1 0

0 ' 0

i

i V

 



 

 

 



 

 n n d

I (

)

2 cos cos

'

: fn. of d and  Ex) A birefringent plate sandwiched between a pair of crossed polarizers











 







 























 

 





 





0 sin2 1

0

cos2 sin2

sin2 cos2

0 0

0

' 1 i

i

i

V

  

 

 



 

 n n d

I (

)

sin

'

Nonlinear Optics Lab

.

Circular polarization representation

It is often more convenient to express the field in terms of “basis” vectors that are circularly polarized ;

















1 : 0 0 CW

: 1

CCW and : constitute a complete set that can be used to describe a field of arbitrary polarization.

Right circularly polarized Left circularly polarized

Rectangular representation : Circular representation :



 







 



 



 



 

y x y

x V

V V

V 1

0 0

V 1







 







 







 







 V

V V

V 1

0 0

V 1